Does anyone know how to mesh this robot claw?
There isn't need for a lot of complex mechanics. In 3D, you often fake it.

This is the fastest way I know how to do it:



Start by making a cylinder in a new scene. I chose these settings at the start. The main thing is, don't give it a cap fill type. I set mine to None.



With your object selected in Object Mode, go to the Modifiers Tab in the Properties Panel and create a Solidify modifier. Change the Thickness setting to 0.5. Then apply the modifier.



In Edit Mode, make sure you're in Face Select Mode (3 key), and box select, then delete the eight faces at the bottom center of the new shape. Then go into Edge Select Mode (2 key) and then Alt + Select an edge at the top center of the same object. Then split these edges by pressing Ctrl + e to bring up the Edge menu, then choose Edge Split. Note: This menu item seems to have been removed in Blender 2.83. If you are using this version, use the search function instead (F3) and type "Edge Split" in the search field to get access to this function).



In Edit Mode, choose Face Select Mode again and press a to select all faces. Press p and from the Separate menu that pops up, choose Loose Parts. Now you have two pieces. If not, then the previous step, where you selected edges and split them, was not completed properly, and some edges remained connected. Go back and check, and repeat that step if necessary.



In Object Mode, select either of these two new claw objects, and go into Edit Mode. Again, select all faces, and this time go to the top of the 3D view, and choose Mesh > Clean Up > Fill Holes. This will automatically cap both ends of your new object. Repeat the above steps for the other object.



Now go into Object Mode and select either claw object, and then Shift select the other claw object. Now go into Edit Mode (since both objects were selected before entering Edit Mode, they are both equally accessible in Edit Mode).  Go into the top orthographic view with Numpad 7, and select all faces again, which will select all faces for both objects. Translate them vertically by pressing g, followed by y, then dragging the mouse upward. Align these parts so that the middle of the bottom gap between these objects is centered on the Worldspace origin (try to get close to it, anyway). You can snap by holding Ctrl while moving the mouse, and get even finer control by additionally holding Shift. When you leave Edit Mode now, both objects will have their object origin centered at this location, which will make hinge movement easier.

 

Go back into Object Mode and, with the 3D cursor at the Worldspace Origin (if it is not there, put it there with Shift + c), press Shift  + a to add a new cylinder. It should appear where the 3D cursor is, which should be the same location as your origin point for the two claw objects (You may find that the Add Cylinder panel remembers some of the settings of our earlier created cylinder). This one needs to be smaller, to fit between the claws, and the Fill Cap Type needs to not be None this time. I chose Ngon, and a radius of 0.2.

Finally, go into Object Mode and select both claw objects, and then add the new cylinder to the selection last. Press Ctrl + p to bring up the Parent Menu, and select Object. The claws will now go where the pivot object goes. If you want, you can lock the axes that shouldn't move in the Transform Panel in the Properties Shelf to the right. You don't even need bones for this setup, nor drivers, really, since you can zero out the rotation values to get your claws back to their starting position, or else press Alt + r to restore the zeroed out values instantly.